Problem: All of the 4th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$35.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$25.50$ each for teachers and $$9.50$ each for students, and the group paid $$108.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 35}$ ${25.5x+9.5y = 108}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-9y = -105}$ ${25.5x+9.5y = 108}$ Add the top and bottom equations together. $ 0.5y = 3 $ $ y = \dfrac{3}{0.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+3y = 35}$ to find $x$ ${8.5x + 3}{(6)}{= 35}$ $8.5x+18 = 35$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {25.5x+9.5y = 108}$ and get the same answer for $x$ ${25.5x + 9.5}{(6)}{= 108}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.